Daily Frieze · One-stop Frieze Group Symmetry

Seven Days of Frieze Group Playalong

This is the one-stop post for making a linear pattern with a repeating tiles, otherwise known as Frieze Group Symmetry.

I recently wrote 10 posts about this kind of pattern making, going over nuances and details one at a time. This post is not that. It’s aim is to demonstrate Frieze Group symmetry in a way that will make the basics of this symmetries really accessible. I hope you will play along. Actually creating the symmetries is the easiest way to learn them. IT IS SO WORTH LEARNING THIS!  Playing with these symmetrie expand design possibilities beyond what you can imagine.

There are many different symmetry groups. Frieze group symmetry is a great place to start because they follow one simple rule: repeated tiles can only move linearly, which means they can slide, thus moving in a linear direction, or they can have 180 degree transformation. There are only 7 possible ways to make a linear symmetry. Considering we generally only think of symmetry as a mirror reflection, this may seem more than you expect.

To play along, you need four squares that

  • have a design on the back that is a reflection of what’s on the front, and
  • which the design on the square is not symmetrical.


The easiest way to do this is to just stack four square post-it notes and  cut an asymmetrical design on them while they are still stuck together.

If you want something a bit more snazzy and have access to a printer, print out either the color or black and white version of my symmetry tiles.

Symmetry tiles to be cut out and folded to make squares.


You’ll cut out four of these rectangles, which are domino proportioned, then fold them in half to make squares. Glue insides together, and you have your own set of paper symmetry tiles to use.

Here are the PDFs

Colorful awesome Symmetry Tile

Awesome Black and White Symmetry Tile

I will be adding to this post for seven days, so we’ll do just one symmetry a day.


If you take your tile keep repeating it exactly how it is, this is called translation and it looks like this:

There you have it! Your first symmetry group.

To be continued on this page tomorrow.

Day 2

Glide Reflection
Glide Reflection

All frieze symmetries have the translation action embedded in them. Here, with Glide Reflection, a copy of the first tile slides over (translation) then it reflects horizontally, as if there is a horizontal bar that it is flipping over. This shows you the back of the tile, which is why the tiles I’ve provided above in my pdfs, give you the image as if it goes right through the paper.

If you are using your own cut squares, be sure that the design you’ve cut makes it so there are no lines of horizontal or vertical symmetry!

If you are confused about this, or want to know more about this particular symmetry, take a look at my glide reflection post.

to be continued tomorrow….

Day 3

I think of this frieze group as the paper doll frieze group. Remember cutting out paper dolls from a piece of paper folded like an accordion, and the result would be linked paper dolls? If you don’t know what I’m talking about, do a google search for accordion paper dolls!

The Vertical Reflection frieze group slides a tile over, then reflects across the right side of the tile. Repeat and repeat again.

For more details, take a look at my Vertical Reflection post.

Day 4

Frieze group, Rotation
Frieze group, Rotation

Some of these symmetries seem easier to wrap my brain around than others. Seems to me that this fourth symmetry, rotation, is one of the easier ones. Remembering that all the symmetries start with translation, which means sliding a copy of the first tile over, this one then rotates 180 degree, going from right side up to upside down.

Day 5

This frieze group combines the first three groups into one. Not sure about glide reflection? Better get clear on it!

Vertical Refection, Glide Reflection
Vertical Refection, Glide Reflection

The action here is to slide over the first tile then reflect it across a vertical axis (mirror reflection).

Now here’s something that’s new to this frieze group: take a copy of the two tiles, slide them over, then reflected them across a horizontal axis. This is glide reflection.

This is the first (of two) of the frieze groups where you need a minimum of four tiles to achieve the symmetry of this specific one of the group.

Day 6

So close to the end of Frieze Group symmetry , here’s one that super easy to understand, Notice that there’s a new thing that happens here.

Horizontal Reflection
Horizontal Reflection

The action is to take a copy of the first tile and rotate it across a horizontal axis, the same kind of reflection that a tree does with the water at it’s base. Next, slide (translate) both these tiles to continue the pattern.

What’s new is that this is the first of these symmetries that require the design to become wider before it becomes longer. In a sense, it happens on double line.

Day 7

If you understand what’s going on in all the other frieze groups, this one will be immediately understandable. If your understanding is still a bit sketchy, you’ll likely need a bit more time to sort it out.

LIke the one from Day 6, this frieze symmetry requires the pattern to grown in wider (or taller, depending on your perspective) before it grows longer.

Horizontally reflect a copy of the first tile. This means reflect the tile the way a tree would reflect into water. Next, reflect both tiles over a vertical line, like a mirror reflection. This is the unit of four repeats linearly.

That’s it! Last of the seven frieze groups. There are lots of nuances to discover to with all of these groups. I’ve  gone over these nuances in the posts that can be accessed with this link https://bookzoompa.wordpress.com/category/frieze-groups/


It’s great to have a reference page to quickly refer to, to remember the different arrangements.

Here’s a PDF of the Frieze Group chart above.

Special shout out to Hana Murray for sharing her play-along photos!


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